Transformation optics has been a major catalyst for the formidable advances
in the field of electromagnetic metamaterials, by providing a systematic and
versatile tool for the conception and design of artificial media with given field-manipulation
capabilities. In essence, by exploiting the form-invariance of Maxwell’s equations with
respect to coordinate transformations, transformation optics allows us to systematically
tailor the spatial constitutive profile of a material so as to precisely manipulate the field distribution
and propagation according to a given coordinate-distorted reference frame.
The implied paradigm shift is the separation of the conceptual design from the actual material synthesis,
with the former essentially driven by geometrical intuition and considerations, and the latter reducing
to a suitable approximation of given ideal constitutive “blueprints.”

It is well known that the inherently magnetic properties of typically arising transformation media
limit the high-frequency scaling of practical metamaterial implementations based on subwavelength inclusions
(e.g., split-ring resonators). In [1, 2], with reference to
the “invisibility cloaking” scenario, we explored
alternative approaches to attain nonmagnetic implementations,
with performance comparable to that of a nonideal lossy, dispersive,
parameter truncated implementation of standard magnetic cloaking.

In [3, 4],
we applied the transformation approach to corner- and wedge-type configurations, in order to attain
unconventional scattering signatures, whereas, in [5], we explored a general
class of transparent metamaterial slabs with intriguing image-displacement/formation capabilities. For instance, the figure
top panels illustrate the design of a coordinate transformation capable of “flattening” a corner-reflector geometry,
thereby yielding a thin planar metamaterial-based retroreflector.
The bottom panels show the numerically computed scattering response for oblique (15°) plane-wave incidence,
comparing the proposed retroreflector to a standard corner reflector and a plain metal sheet.
The retrodirective response of the proposed reﬂector is clearly visible from the wavefront shapes,
in complete accord with those observed in the corner-reﬂector case, and in sharp contrast with
those observed in the plain-metal-sheet (which instead exhibits a strong specular response).

In [6], we studied the interactions between an invisibility cloak
and an “anti-cloak” capable of (partially or totally) “undoing” the cloaking transformation.
In [7, 8], we further developed this concept,
and illustrated its possible applications to the design of “invisible” sensors.

In [9], we extended the approach to
deal with single-negative materials, and applied this concept to the design of
tunneling effects.

More recently, we have applied the transformation-optics approach to the design of metamaterial radomes for extending
the scanning angle of a phased array [10].

Relevant papers

In coordinate-transformation-based approaches to electromagnetic concealment (“cloaking”) of objects, use of higher-order (quadratic) mappings has been proposed as an effective device to obtain satisfactory responses without the use of magnetic materials that are complicated to synthesize at optical frequencies. In this article, we explore a new higher-order algebraic transformation, which allows, in principle, for a broader range of applicability and further parametric optimization. Via full-wave numerical studies of near- and far-field observables, we assess its performance by comparison with various reference cases (nonreduced parameters, quadratic transformation, no cloak).

Coordinate-transformation approaches to invisibility cloaking rely on the design of an anisotropic, spatially inhomogeneous “transformation medium” capable of suitably rerouting the energy flux around the region to conceal without causing any scattering in the exterior region. It is well known that the inherently magnetic properties of such medium limit the high-frequency scaling of practical “metamaterial” implementations based on subwavelength inclusions (e.g., split-ring resonators). Thus, for the optical range, nonmagnetic implementations, based on approximate reductions of the constitutive parameters, have been proposed. In this paper, we present an alternative approach to nonmagnetic coordinate-transformation cloaking, based on the mapping from a nearly transparent, anisotropic and spatially inhomogeneous virtual domain. We show that, unlike its counterparts in the literature, our approach is amenable to exact analytic treatment, and that its overall performance is comparable to that of a nonideal (lossy, dispersive, parameter truncated) implementation of standard (magnetic) cloaking.

In this letter, we address the design of thin planar retrodirective reflectors via transformation optics. Exploiting form-invariant transformations of Maxwell’s equations, we derive the constitutive properties of an anisotropic and spatially inhomogeneous transformation-medium coating which, laid on a flat metallic surface, exhibits the retrodirective response typical of a dihedral corner reflector. The practical feasibility of the involved materials should be within reach of the fast-pacing metamaterial technology. The results of our investigations, validated via a full-wave study of the near- and far-field responses, could find interesting applications in radar, communication, and identification scenarios.

Transformation optics has recently emerged as a powerful and systematic approach to design application-oriented metamaterials. In this letter, following up on our previous studies on thin planar retroreflectors, we show how it is possible, in principle, to design “transformation medium” coatings capable of controlling the scattering response of metallic corner- and wedge-type structures so as, e.g., to strongly enhance the specularly reflected component. We validate our results via a full-wave study of the near- and far-field responses, and envisage possible applications.

In this paper, we apply transformation-based optics to the derivation of a general class of transparent metamaterial slabs. By means of analytical and numerical full-wave studies, we explore their image-displacement/formation capabilities, and establish intriguing connections with configurations already known in the literature. Starting from these revisitations, we develop a number of nontrivial extensions, and illustrate their possible applications to the design of perfect radomes, anticloaking devices, and focusing devices based on double-positive (possibly nonmagnetic) media. These designs show that such anomalous features may be achieved without necessarily relying on negative-index or strongly resonant metamaterials, suggesting more practical venues for the realization of these devices.

Coordinate-transformation cloaking is based on the design of a metamaterial shell made of an anisotropic, spatially inhomogeneous “transformation medium” that allows rerouting the impinging wave around a given region of space. In its original version, it is generally believed that, in the ideal limit, the radiation cannot penetrate the cloaking shell (from outside to inside, and viceversa). However, it was recently shown by Chen et al. that electromagnetic fields may actually penetrate the cloaked region, provided that this region contains double-negative transformation media which, via proper design, may be in principle used to (partially or totally) “undo” the cloaking transformation, thereby acting as an “anti-cloak.” In this paper, we further elaborate this concept, by considering a more general scenario of cloak/anti-cloak interactions. Our full-wave analytical study provides new insightful results and explores the effects of departure from ideality, suggesting also some novel scenarios for potential applications.

The suggestive idea of “cloaking” an electromagnetic sensor, i.e., strongly reducing its visibility (scattering) while maintaining its field-sensing (absorption) capabilities, has recently been proposed in the literature, based on scattering-cancellation, Fano-resonance, or transformation-optics approaches. In this paper, we explore an alternative transformation-optics-based route, which relies on the recently introduced concept of “anti-cloaking.” More specifically, our proposed approach relies on a suitable tailoring of the competing cloaking and anti-cloaking mechanisms, interacting in a two-dimensional cylindrical scenario. Via analytical and parametric studies, we illustrate the underlying phenomenology, identify the critical design parameters, and address the relevant optimality and trade-off issues, taking also into account the effect of material losses. Our results confirm the envisaged potentials of the proposed transformation-optics approach as an attractive alternative route to sensor cloaking.

The intriguing concept of “anti-cloaking” has been recently introduced within the framework of transformation optics (TO), first as a “countermeasure” to invisibility-cloaking (i.e., to restore the scattering response of a cloaked target), and more recently in connection with “sensor invisibility” (i.e., to strongly reduce the scattering response while maintaining the field-sensing capabilities). In this paper, we extend our previous studies, which were limited to a two-dimensional cylindrical scenario, to the three-dimensional spherical case. More specifically, via a generalized (coordinate-mapped) Mie-series approach, we derive a general analytical full-wave solution pertaining to plane-wave-excited configurations featuring a spherical object surrounded by a TO-based invisibility cloak coupled via a vacuum layer to an anti-cloak, and explore the various interactions of interest. With a number of selected examples, we illustrate the cloaking and field-restoring capabilities of various configurations, highlighting similarities and differences with respect to the cylindrical case, with special emphasis on sensor-cloaking scenarios and ideas for approximate implementations that require the use of double-positive media only.

Transformation media designed by standard transformation-optics (TO) approaches, based on real-valued coordinate mapping, cannot exhibit single-negative (SNG) character unless such character is already possessed by the domain that is being transformed. In this paper, we show that, for a given field polarization, pseudo-SNG transformation media can be obtained by transforming a domain featuring double positive (or double-negative) character, via complex analytic continuation of the coordinate transformation rules. Moreover, we apply this concept to the TO-based interpretation of phenomena analogous to the tunnelling effects observable in bi-layers made of complementary epsilon-negative (ENG) and mu-negative (MNG) media, and explore their possible TO-inspired extensions and generalizations.

We apply the transformation-optics approach to the design of a metamaterial radome that can extend the scanning angle of a phased-array antenna. For moderate enhancement of the scanning angle, via suitable parameterization and optimization of the coordinate transformation, we obtain a design that admits a technologically viable, robust and potentially broadband implementation in terms of thin-metallic-plate inclusions. Our results, validated via finite-element-based numerical simulations, indicate an alternative route to the design of metamaterial radomes which does not require negative-valued and/or extreme constitutive parameters.

Transformation optics asks, using Maxwell’s equations, what kind of electromagnetic medium recreates some smooth deformation of space? The guiding principle is Einstein’s principle of covariance: that any physical theory must take the same form in any coordinate system. This requirement fixes very precisely the required electromagnetic medium. The impact of this insight cannot be overestimated. Many practitioners were used to thinking that only a few analytic solutions to Maxwell’s equations existed, such as the monochromatic plane wave in a homogeneous, isotropic medium. At a stroke, transformation optics increases that landscape from ‘few’ to ‘infinity’, and to each of the infinitude of analytic solutions dreamt up by the researcher, there corresponds an electromagnetic medium capable of reproducing that solution precisely. The most striking example is the electromagnetic cloak, thought to be an unreachable dream of science fiction writers, but realised in the laboratory a few months after the papers proposing the possibility were published. But the practical challenges are considerable, requiring meta-media that are at once electrically and magnetically inhomogeneous and anisotropic. How far have we come since the first demonstrations over a decade ago? And what does the future hold? If the wizardry of perfect macroscopic optical invisibility still eludes us in practice, then what compromises still enable us to create interesting, useful, devices? While three-dimensional (3D) cloaking remains a significant technical challenge, much progress has been made in two dimensions. Carpet cloaking, wherein an object is hidden under a surface that appears optically flat, relaxes the constraints of extreme electromagnetic parameters. Surface wave cloaking guides sub-wavelength surface waves, making uneven surfaces appear flat. Two dimensions is also the setting in which conformal and complex coordinate transformations are realisable, and the possibilities in this restricted domain do not appear to have been exhausted yet. Beyond cloaking, the enhanced electromagnetic landscape provided by transformation optics has shown how fully analytic solutions can be found to a number of physical scenarios such as plasmonic systems used in electron energy loss spectroscopy and cathodoluminescence. Are there further fields to be enriched? A new twist to transformation optics was the extension to the spacetime domain. By applying transformations to spacetime, rather than just space, it was shown that events rather than objects could be hidden from view; transformation optics had provided a means of effectively redacting events from history. The hype quickly settled into serious nonlinear optical experiments that demonstrated the soundness of the idea, and it is now possible to consider the practical implications, particularly in optical signal processing, of having an ‘interrupt-without-interrupt’ facility that the so-called temporal cloak provides. Inevitable issues of dispersion in actual systems have only begun to be addressed. Now that time is included in the programme of transformation optics, it is natural to ask what role ideas from general relativity can play in shaping the future of transformation optics. Indeed, one of the earliest papers on transformation optics was provocatively titled ‘General Relativity in Electrical Engineering’. The answer that curvature does not enter directly into transformation optics merely encourages us to speculate on the role of transformation optics in defining laboratory analogues. Quite why Maxwell’s theory defines a ‘perfect’ transformation theory, while other areas of physics such as acoustics are not apparently quite so amenable, is a deep question whose precise, mathematical answer will help inform us of the extent to which similar ideas can be extended to other fields. The contributors to this Roadmap, who are all renowned practitioners or inventors of transformation optics, will give their perspectives into the field’s status and future development.